On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$

  • N. V.  Krylov  Univ. Minnesota, Minneapolis, MN, USA
Keywords: Itˆo’s equations with singular drift, Markov diffusion processes


UDC 519.21

We prove the solvability of Itô stochastic equations with uniformly nondegenerate bounded measurable diffusion and drift in $L_{d+1}(R^{d+1}).$
Actually, the powers of summability of the drift in $x$ and $t$ could be different.
Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A. V. Skorokhod.
Weak uniqueness of solutions is an open problem even if the diffusion is constant.

Author Biography

N. V.  Krylov,  Univ. Minnesota, Minneapolis, MN, USA

 Univ. Minnesota, Minneapolis, MN, USA


S. V. Anulova, G. Pragarauskas, Weak Markov solutions of stochastic equations, Litovsk. Mat. Sb., 17, No. 2, 5 – 26 (1977); English translation: Lith. Math. J., 17, No. 2, 141 – 155 (1977)

L. Beck, F. Flandoli, M. Gubinelli, M. Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift:

regularity, duality and uniqueness, Electron. J. Probab., 24, No. 136, 1 – 72 (2019), https://doi.org/10.1214/19-ejp379

E. B. Dynkin, Markov processes, Fizmatgiz, Moscow (1963); English translation: Grundlehren Math. Wiss., Vols. 121, 122, Springer-Verlag, Berlin (1965).

N. V. Krylov, On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes, Izv. Akad. Nauk SSSR, ser. mat., 37, No. 3, 691 – 708 (1973); English translation: Math. USSR Izv., 7, No. 3, 691 – 709 (1973).

N. V. Krylov, Controlled diffusion processes, Nauka, Moscow (1977); English translation: Springer (1980). xii+308 pp. ISBN: 0-387-90461-1

N. V. Krylov, On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale, Mat. Sb., 130, No. 2, 207 – 221 (1986); English translation: Math. USSR Sb., 58, No. 1, 207 – 222(1987), https://doi.org/10.1070/SM1987v058n01ABEH003100

N. V. Krylov, Introduction to the theory of diffusion processes, Amer. Math. Soc., Providence, RI (1995). xii+271 pp. ISBN: 0-8218-4600-0, https://doi.org/10.1090/mmono/142

N. V. Krylov, Sobolev and viscosity solutions for fully nonlinear elliptic and parabolic equations, Math. Surveys and Monogr., 233, Amer. Math. Soc., Providence, RI (2018). xiv+441 pp. ISBN: 978-1-4704-4740-3, https://doi.org/10.1090/surv/233

N. V. Krylov, On stochastic equations with drift in $L_d$ ; http://arxiv.org/abs/2001.04008.

Kyeongsik Nam, Stochastic differential equations with critical drifts, arXiv:1802.00074 (2018).

A. I. Nazarov, Interpolation of linear spaces and estimates for the maximum of a solution for parabolic equations, Partial Different. Equat., Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk (1987), 50 – 72; Translated into English as On the maximum principle for parabolic equations with unbounded coefficients, https:// arxiv.org/abs/1507.05232.

N. I. Portenko, Generalized diffusion processes, Nauka, Moscow (1982): English translation: Amer. Math. Soc., Providence, Rhode Island (1990). x+180 pp. ISBN: 0-8218-4538-1, https://doi.org/10.1090/mmono/083

A. V. Skorokhod, Studies in the theory of random processes, Kiev Univ. Press (1961); English translation: Scripta Technica, Washington (1965).

D. W. Stroock, S. R. S. Varadhan, Multidimensional diffusion processes, Grundlehren Math. Wiss., 233, Berlin, New York, Springer-Verlag (1979).

Longjie Xie, Xicheng Zhang, Ergodicity of stochastic differential equations with jumps and singular coefficients, Ann. Inst. Poincare Probab. Stat., ´ 56, No. 1, 175 – 229 (2020), https://doi.org/10.1214/19-AIHP959

T. Yastrzhembskiy, A note on the strong Feller property of diffusion processes; arXiv:2001.09919.

I. Gyöngy, T. Martínez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J., 51(126), No 4, 763 – 783 (2001), https://doi.org/10.1023/A:1013764929351

How to Cite
KrylovN. V. “On Time Inhomogeneous Stochastic Itô Equations With Drift in $L_{d+1}$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 9, Sept. 2020, pp. 1232-53, doi:10.37863/umzh.v72i9.6280.
Research articles